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Pythagoras' Theorem

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Pythagoras' Theorem allows the length of one side of a triangle to be calculated if the lengths of the other two sides are known. This is especially useful when finding the distance between two points of a coordinate system.

Pythagoras and his theorem

Pythagoras was a Greek mathematician and philosopher who lived over 2500 years ago. He is often credited with discovering the relationship between the length of the sides on a right-angled triangle although this relationship was used by others long before Pythagoras' time.

Work through guidance below on Pythagoras' Theorem (a.k.a. Pythagorean Theorem) with your children and then print out the worksheets listed toward the bottom of this page and have them try the questions.

The right-angled triangle below is shown along with the squares of the length of each of its three sides.

right triangle with the squares of each side shown

The  area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

c2 = a 2 + b 2

Pythagoras' Theorem is, that for a right-angled triangle, the area of the square on the hypotenuse (the hypotenuse is the longest side in a right-angled triangle) equals the sum of the areas of the squares on the other two sides . In other words, the square of the longest side equals the squares of the two other sides added together.

Make sure your children are familiar with the terminology and concept around squaring numbers (multiplying a number by itself). e.g. 4 squared, or 42 = 4 x 4 = 16. You will find more here on squares and square roots.

Pythagoras Proof

There are many different proofs of Pythagoras' Theorem. You will find a very good listing hereexternal resource icon. Note: You would usually only be expected to recall one of these proofs.

Examples of Solving Problems with Pythagoras' Theorem

right triangle with sides of length 3 and 4 and hypotenuse unknown

What is the length of side c?

Recall formula c2 = a2 + b2
Substitute in numbers c2 = 32 + 42
Note: Since we will be adding a2 and b2 it does not matter which side is "a" and which is "b". Just call one "a" and the other "b".
Square both numbers 32 = 3 x 3 = 9
42 = 4 x 4 = 16
Add them together c2 = 9 + 16
c2 = 25
Find the square root (the number that multiplied by itself equals 25)

c = √25

c = 5

 

More Problems Solved With Pythagoras

right triangle with sides of length 6 and 8 and hypotenuse unknown

c2 = 62 + 82

c2 = 36 + 64

c2 = 100

c = √100

c = 10

 

right triangle with sides of length 5 and 12 and hypotenuse unknown

c2 = 52 + 122

c2 = 25 + 144

c2 = 169

c = √169

c = 13

 

right triangle with sides of length 7 and 9 and hypotenuse unknown

c2 = 72 + 92

c2 = 49 + 81

c2 = 130

c = √130

c = 11.40

 

Two points to remember when checking your answers:

  • the hypotenuse (the side opposite the right angle) is always the longest of the three sides
  • the hypotenuse can never be longer than the length of the other two sides combined.

More problems with other sides unknown

Pythagoras' Theorem can be used to calculate the length of any side of a right-angled triangle if the other two lengths are known.

right triangle with hypotenuse of 20, a side of length 16, and an unknown side

202 = a2 + 162

202 - 162 = a2 + 162 - 162

400 - 256 = a2

144 = a2

√144 = a

a = 12

right triangle with hypotenuse of 20, a side of length 16, and an unknown side

122 = 9 2 + b2

122 - 9 2 = 92 - 92 + b2

144 - 81 = b 2

63 = b2

√63 = b

b = 7.94

When the hypotenuse is one of the two known lengths, as in the two examples above, the shorter length is squared and then subtracted from the square of the hypotenuse.

Pythagorean Triangle

You may have noticed that in some of the examples above, the lengths of each side were all whole numbers. These triangles are known as Pythagorean Triangles. The best known of these is the 3, 4, 5 triangle. There are other triangles that have all three sides with an exact whole number for their lengths. These include triangles with lengths of sides ( 5, 12, 13) , ( 7, 24, 25) , & ( 8, 15, 17).

Pythagoras' Theorem Worksheets

Print out the Pythagoras' Theorem worksheets below and have your children try the questions.

Applying Pythagoras' Theorem

There are examples here showing how Pythagoras' Theorem can be applied to find the distance between two points with given coordinates.

 

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